1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Beta</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../sf_beta.html" title="Beta Functions"> 9<link rel="prev" href="../sf_beta.html" title="Beta Functions"> 10<link rel="next" href="ibeta_function.html" title="Incomplete Beta Functions"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../sf_beta.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ibeta_function.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.sf_beta.beta_function"></a><a class="link" href="beta_function.html" title="Beta">Beta</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.sf_beta.beta_function.h0"></a> 31 <span class="phrase"><a name="math_toolkit.sf_beta.beta_function.synopsis"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.synopsis">Synopsis</a> 32 </h5> 33<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34</pre> 35<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> 36 37<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> 38<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span> 39 40<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 41<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 42 43<span class="special">}}</span> <span class="comment">// namespaces</span> 44</pre> 45<h5> 46<a name="math_toolkit.sf_beta.beta_function.h1"></a> 47 <span class="phrase"><a name="math_toolkit.sf_beta.beta_function.description"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.description">Description</a> 48 </h5> 49<p> 50 The beta function is defined by: 51 </p> 52<div class="blockquote"><blockquote class="blockquote"><p> 53 <span class="inlinemediaobject"><img src="../../../equations/beta1.svg"></span> 54 55 </p></blockquote></div> 56<div class="blockquote"><blockquote class="blockquote"><p> 57 <span class="inlinemediaobject"><img src="../../../graphs/beta.svg" align="middle"></span> 58 59 </p></blockquote></div> 60<p> 61 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 62 be used to control the behaviour of the function: how it handles errors, 63 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 64 documentation for more details</a>. 65 </p> 66<p> 67 The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result 68 type calculation rules</em></span></a> when T1 and T2 are different types. 69 </p> 70<h5> 71<a name="math_toolkit.sf_beta.beta_function.h2"></a> 72 <span class="phrase"><a name="math_toolkit.sf_beta.beta_function.accuracy"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.accuracy">Accuracy</a> 73 </h5> 74<p> 75 The following table shows peak errors for various domains of input arguments, 76 along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> 77 and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. 78 Note that only results for the widest floating point type on the system are 79 given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 80 zero error</a>. 81 </p> 82<div class="table"> 83<a name="math_toolkit.sf_beta.beta_function.table_beta"></a><p class="title"><b>Table 8.17. Error rates for beta</b></p> 84<div class="table-contents"><table class="table" summary="Error rates for beta"> 85<colgroup> 86<col> 87<col> 88<col> 89<col> 90<col> 91</colgroup> 92<thead><tr> 93<th> 94 </th> 95<th> 96 <p> 97 GNU C++ version 7.1.0<br> linux<br> double 98 </p> 99 </th> 100<th> 101 <p> 102 GNU C++ version 7.1.0<br> linux<br> long double 103 </p> 104 </th> 105<th> 106 <p> 107 Sun compiler version 0x5150<br> Sun Solaris<br> long double 108 </p> 109 </th> 110<th> 111 <p> 112 Microsoft Visual C++ version 14.1<br> Win32<br> double 113 </p> 114 </th> 115</tr></thead> 116<tbody> 117<tr> 118<td> 119 <p> 120 Beta Function: Small Values 121 </p> 122 </td> 123<td> 124 <p> 125 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 126 2.1:</em></span> <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_beta_GSL_2_1_Beta_Function_Small_Values">And 127 other failures.</a>)</span><br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> 128 Max = 1.14ε (Mean = 0.574ε)) 129 </p> 130 </td> 131<td> 132 <p> 133 <span class="blue">Max = 2.86ε (Mean = 1.22ε)</span><br> <br> 134 (<span class="emphasis"><em><cmath>:</em></span> Max = 364ε (Mean = 76.2ε)) 135 </p> 136 </td> 137<td> 138 <p> 139 <span class="blue">Max = 2.86ε (Mean = 1.22ε)</span> 140 </p> 141 </td> 142<td> 143 <p> 144 <span class="blue">Max = 2.23ε (Mean = 1.14ε)</span> 145 </p> 146 </td> 147</tr> 148<tr> 149<td> 150 <p> 151 Beta Function: Medium Values 152 </p> 153 </td> 154<td> 155 <p> 156 <span class="blue">Max = 0.978ε (Mean = 0.0595ε)</span><br> 157 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.18e+03ε (Mean = 238ε))<br> 158 (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.09e+03ε (Mean = 265ε)) 159 </p> 160 </td> 161<td> 162 <p> 163 <span class="blue">Max = 61.4ε (Mean = 19.4ε)</span><br> <br> 164 (<span class="emphasis"><em><cmath>:</em></span> Max = 1.07e+03ε (Mean = 264ε)) 165 </p> 166 </td> 167<td> 168 <p> 169 <span class="blue">Max = 107ε (Mean = 24.5ε)</span> 170 </p> 171 </td> 172<td> 173 <p> 174 <span class="blue">Max = 96.5ε (Mean = 22.4ε)</span> 175 </p> 176 </td> 177</tr> 178<tr> 179<td> 180 <p> 181 Beta Function: Divergent Values 182 </p> 183 </td> 184<td> 185 <p> 186 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 187 2.1:</em></span> Max = 12.1ε (Mean = 1.99ε))<br> (<span class="emphasis"><em>Rmath 188 3.2.3:</em></span> Max = 176ε (Mean = 28ε)) 189 </p> 190 </td> 191<td> 192 <p> 193 <span class="blue">Max = 8.99ε (Mean = 2.44ε)</span><br> <br> 194 (<span class="emphasis"><em><cmath>:</em></span> Max = 128ε (Mean = 23.8ε)) 195 </p> 196 </td> 197<td> 198 <p> 199 <span class="blue">Max = 18.8ε (Mean = 2.71ε)</span> 200 </p> 201 </td> 202<td> 203 <p> 204 <span class="blue">Max = 11.4ε (Mean = 2.19ε)</span> 205 </p> 206 </td> 207</tr> 208</tbody> 209</table></div> 210</div> 211<br class="table-break"><p> 212 Note that the worst errors occur when a or b are large, and that when this 213 is the case the result is very close to zero, so absolute errors will be 214 very small. 215 </p> 216<h5> 217<a name="math_toolkit.sf_beta.beta_function.h3"></a> 218 <span class="phrase"><a name="math_toolkit.sf_beta.beta_function.testing"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.testing">Testing</a> 219 </h5> 220<p> 221 A mixture of spot tests of exact values, and randomly generated test data 222 are used: the test data was computed using <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> 223 at 1000-bit precision. 224 </p> 225<h5> 226<a name="math_toolkit.sf_beta.beta_function.h4"></a> 227 <span class="phrase"><a name="math_toolkit.sf_beta.beta_function.implementation"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.implementation">Implementation</a> 228 </h5> 229<p> 230 Traditional methods of evaluating the beta function either involve evaluating 231 the gamma functions directly, or taking logarithms and then exponentiating 232 the result. However, the former is prone to overflows for even very modest 233 arguments, while the latter is prone to cancellation errors. As an alternative, 234 if we regard the gamma function as a white-box containing the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos 235 approximation</a>, then we can combine the power terms: 236 </p> 237<div class="blockquote"><blockquote class="blockquote"><p> 238 <span class="inlinemediaobject"><img src="../../../equations/beta2.svg"></span> 239 240 </p></blockquote></div> 241<p> 242 which is almost the ideal solution, however almost all of the error occurs 243 in evaluating the power terms when <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> 244 are large. If we assume that <span class="emphasis"><em>a > b</em></span> then the larger 245 of the two power terms can be reduced by a factor of <span class="emphasis"><em>b</em></span>, 246 which immediately cuts the maximum error in half: 247 </p> 248<div class="blockquote"><blockquote class="blockquote"><p> 249 <span class="inlinemediaobject"><img src="../../../equations/beta3.svg"></span> 250 251 </p></blockquote></div> 252<p> 253 This may not be the final solution, but it is very competitive compared to 254 other implementation methods. 255 </p> 256<p> 257 The generic implementation - where no <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos 258 approximation</a> approximation is available - is implemented in a very 259 similar way to the generic version of the gamma function by means of Sterling's 260 approximation. Again in order to avoid numerical overflow the power terms 261 that prefix the series are collected together 262 </p> 263<p> 264 There are a few special cases worth mentioning: 265 </p> 266<p> 267 When <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> are less than one, 268 we can use the recurrence relations: 269 </p> 270<div class="blockquote"><blockquote class="blockquote"><p> 271 <span class="inlinemediaobject"><img src="../../../equations/beta4.svg"></span> 272 273 </p></blockquote></div> 274<div class="blockquote"><blockquote class="blockquote"><p> 275 <span class="inlinemediaobject"><img src="../../../equations/beta5.svg"></span> 276 277 </p></blockquote></div> 278<p> 279 to move to a more favorable region where they are both greater than 1. 280 </p> 281<p> 282 In addition: 283 </p> 284<div class="blockquote"><blockquote class="blockquote"><p> 285 <span class="inlinemediaobject"><img src="../../../equations/beta7.svg"></span> 286 287 </p></blockquote></div> 288</div> 289<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 290<td align="left"></td> 291<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 292 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 293 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 294 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 295 Daryle Walker and Xiaogang Zhang<p> 296 Distributed under the Boost Software License, Version 1.0. (See accompanying 297 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 298 </p> 299</div></td> 300</tr></table> 301<hr> 302<div class="spirit-nav"> 303<a accesskey="p" href="../sf_beta.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ibeta_function.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 304</div> 305</body> 306</html> 307
